T O measure a nation's wealth, an industry's productive potential or the consumption of a durable stock, we must be able to add machines with different characteristics and different vintages to form an aggregate. Ideally, such a measure would change with machinery deterioration and obsolescence but not with pure price level changes which leave the use of the machinery the same. The aggregation task would be easier if we had information about the nature of depreciation. For example, if depreciation is a constant rate, and that rate remains the same over time, then the aggregate is a simple weighted average of the component machines, the weights being derived from the known depreciation rate. This model is not uncommon, yet its assumptions are clearly restrictive. It would be very useful if there were sufficient empirical evidence pertaining to the nature of depreciation patterns to either confirm or reject such a simple model. It is the intent of this paper to provide some of that evidence. One natural approach to studying decay of capital is to study the in-use cost of machines as they age. Depreciation values could be estimated from changes in rental prices throughout a machine's life. In the absence of welldeveloped rental markets, however, resale values would yield approximations of the remaining value of machinery after a period of use. This paper constructs actual depreciation figures for automobiles from purchase prices, and studies assumptions and hypotheses about the relationship between new and used machinery. In particular, three assumptions are common. First, it is often assumed that depreciation patterns remain fixed over time. For this assumption to be valid, any technological change must be either nonexistent or smooth. There can be no sudden, dramatic innovations, since these would change the nature of depreciation schemes. Similarly, it is often assumed that machinery of the same type depreciates in the same fashion. This assumption will also be studied. The third and most common assumption is that equipment depreciates at a constant rate.' This is a very useful assumption, since it greatly simplifies the relationship between new and used pieces of equipment. These assumptions will be tested for automobiles using figures for nineteen different makes from 1950 to 1969.

OST models of portfolio selection have M been one-period models. I examine the combined problem of optimal portfolio selection and consumption rules for an individual in a continuous-time model whzere his income is generated by returns on assets and these returns or instantaneous growth rates are stochastic. P. A. Samuelson has developed a similar model in discrete-time for more general probability distributions in a companion paper [8]. I derive the optimality equations for a multiasset problem when the rate of returns are generated by a Wiener Brownian-motion process. A particular case examined in detail is the two-asset model with constant relative riskaversion or iso-elastic marginal utility. An explicit solution is also found for the case of constant absolute risk-aversion. The general technique employed can be used to examine a wide class of intertemporal economic problems under uncertainty. In addition to the Samuelson paper [8], there is the multi-period analysis of Tobin [9]. Phelps [6] has a model used to determine the optimal consumption rule for a multi-period example where income is partly generated by an asset with an uncertain return. Mirrless [5] has developed a continuous-time optimal consumption model of the neoclassical type with technical progress a random variable.

that the size coefficient (i8jl) was statistically significant at the 0.05 level in thirty-five of the 118 industries examined.6 The results for these industries are reported in table 1. The range of variation in the magnitude of i,/3 is considerable; the high, in Perfume and Cosmetics, is 0.114, and the low, 0.014, is found in Men's Clothing. The mean coefficient for the thirty-five industries is 0.0405 suggesting that on the basis of this group one would expect the rate of return to rise by 1.2 per cent if size doubles, and by 4.05 per cent for a ten-fold increase in size. VI Conclusions

M ODERN investment functions, springing from the work of Jorgenson,' differ from earlier investment functions in that they start with an explicit assumption about the economy's aggregate production function. In particular, Jorgenson assumes a Cobb-Douglas production function. Starting with an explicit production function means that it is possible to calculate algebraically the impact of factors, such as interest rates, that could not be isolated in earlier investment functions. Choosing the correct production function is important in estimating partial effects, but the proper definition of the cost of capital variable is also central to their correct estimation. Formulations other than those of Jorgenson are possible. If one had priors about the differences in the opportunity cost of capital under the Duesenberry supply of funds hypothesis,2 the cost of capital could be defined to embody these priors. Doing so would lead to different estimates of the partial effects of tax rates, interest rates, and depreciation policies. Thus, the partial effects that emerge from a modern investment function are a product of the initial specifications of the production function and the cost of capital variable. In addition to choosing the correct production function and the correct definition of the cost of capital, there are other directions in which the modern investment function can be modified. In Jorgenson's neoclassical equilibrium world the cost of capital and the marginal product of capital are always identical. Thus, the desired capital stock at any moment of time is equal to output divided by the marginal product of capital (the cost of capital) multiplied by the elasticity of output with respect to capital. Thus, the only problems are ones of correct data measurement and estimation of the lag structure. This formulation has some theoretical problems. Introducing lags means that the economy is not in equilibrium. actual capital stock lags behind the desired capital stock. Therefore, the cost of capital and the marginal product of capital are not equal. Even if they were equal, the marginal product of capital will differ before and after expansion of the capital stock. Thus output should be divided by the expected cost of capital rather than the actual cost of capital to determine the desired capital stock.3 In a disequilibrium world, the cost of capital and the marginal product of capital can diverge. Profit maximizing firms invest to eliminate the gap between the marginal product of capital and the cost of capital. investment necessary to eliminate this gap depends upon the economy's production function. This paper investigates a disequilibrium investment function based on a Cobb-Douglas production function and Jorgenson's definition of the cost of capital. I was led to investigate such a model in the process of attempting to use the Jorgenson investment function.4 Several problems emerged in addition to those investigated elsewhere.5 (1) Although the Jorgenson investment function fit quarterly time series data for producers' * author would like to thank the referee for many useful comments. 'Dale W. Jorgenson, Anticipations and Behavior, in J. S. Duesenberry, E. Kuh, G. Fromm, and L. R. Klein (editors), Brookings Quarterly Econometric Model of the United States (Chicago: Rand McNally, 1965). Rational Distributed Lag Functions, Econometrica, XXXIV (Jan. 1966), 135-149. With Calvin D. Siebert, A Comparison of Alternative Theories of Corporate Behavior, American Economic Review, XVIII (Sept. 1968). Optimal Capital Accumulation and Corporate Behavior, Journal of Political Economy, LXXVI (Nov./Dec. 1968), 1123-1151. With J. A. Stephenson, The Time Structure of Behavior in United States Manufacturing, 1947-60, this REvIEw, XLIV (Feb. 1967), 16-27. Investment Behavior in U.S. Manufacturing, 1947-60, Econometrica, XXXV (April 1967), 169-220. 2J. Duesenberry, Business Cycles and Economic Growth (New York: McGraw-Hill, 1968), 87-112. 3This was pointed out to me by my colleague Duncan Foley. 'Anyone wishing the detailed econometric results of my attempts to fit the Jorgenson model to producer's durable equipment and nonresidential structures can have them by writing to me. 'Robert Eisner and M. I. Nadiri, Investment Behavior and Neoclassical Theory, this REvIEw, L (Aug. 1968).

D IFFERENTIATING a production function with respect to capital and labor yields equations for the marginal productivity of capital and the marginal productivity of labor. The variables that determine the marginal products in these equations are the same as those in the production function. If the data used to estimate the parameters of the production function are inserted into the equations for the marginal products, the marginal productivities of both capital and labor can be estimated empirically. The same data and equations also make it possible to determine the causes of any changes in the marginal products. How much of the rising marginal productivity of labor is caused by technical progress; how much is caused by a rising capital-labor ratio? If the economy is in equilibrium and there are no economies or diseconomies of scale, actual and marginal returns should be identical. Any differences between the estimated marginal products and the actual returns to capital and labor means that the economy is in disequilibrium; the size of the differences measures the extent of disequilibrium. If disequilibrium does exist, what causes the observed pattern? What are its implications for investment decisions in both human and physical capital? This paper applies the above approach to the American economy from 1929 to 1965.

F LUCTUATIONS in economic activity depend not only on variation in investment but also on variation in saving. How flexible or inflexible saving will be depends partially on previous commitments to save. The concept of saving, has been defined ' by the Survey Research Center of the University of Michigan to include life insurance premium payments,2 payments into retirement or pension funds,3 and principal payments on mortgage debt.4 Non-contractual forms of saving may be called discretionary. All of the components included in the Survey Research Center's definition have common characteristics: (1) Each hinges on a previous contract limiting the possibility of spending on consumer goods. (2) The contractual commitment is of a long-term nature. Two factors would seem to make contractual saving relatively stable: (1) the habitual response to previous decisions, and (2) the economic loss suffered by prematurely discontinuing a long-term commitment to save. The primary purpose of this study is to search for the factors influencing the level of contractual saving. Tests of significance are applied by comparing the coefficients of the selected variables in multiple regressions with their respective standard errors. Our work is different and more comprehensive than the analysis of life insurance premiums by Professors Kreinin. Lansing. and Morgan.5 We are concerned with the aggregate amount of contractual saving rather than with only one of its components. Also, the population is divided into a low-income group and la high-income group with separate analyses for each. The analysis of contractual saving is becoming increasingly important since the trend of contractual saving as a per cent of total saving is upward. In 1949, contractual saving was almost as large as total saving since discretionary dissaving largely offset positive discretionary saving. In explaining aggregate contractual saving both for the high-income group and for the low-income group, our basic theory was clearly confirmed. Disposable income, marital status and the presence or absence of children, and educational level all proved to be significant.6 The power of their influence was in the order stated with disposable income dominant. Race did prove to be significant for the high-income group while age was significant within the lowincome group. Further details must wait.

T HIS paper reports on a study of the relation between peoples' incomes and the way they allocate their time to income and leisure. Time can be spent either on the earning of income (work) or on a host of alternative noneconomic activities (leisure). It is a question of real significance whether, as incomes rise, people systematically change their distribution of time between these activities. However, it is a question to which economics provides no agreed upon answer. No a priori or theoretical answer is possible because we know that, as sellers of their own time, people are pulled in opposite directions by conflicting income and substitution effects. On the other hand, no empirical evidence has thus far been widely accepted despite surprisingly consistent empirical results. But since economic growth and rising incomes are part of a pervasive economic climate, this aspect of peoples' behavior the way they respond to increased economic well-being is of very real importance, in part because it will feed back as an influence on the rate and pattern of growth itself, both in advanced and in underdeveloDed countries.' This subject, of course, is discussed in the theory of public finance as the incentive effect of tax and expenditure, in macro and labor theory as the shape of the long-run aggregate labor supply curve, and in the literature of economic development as the response to changing sectoral terms of trade. This study is empirical. It uses aggregate international cross-sectional data to reveal the typical relation between incomes and hours of work-i.e., between incomes and the allocation of effort (as synonymous with time) to the acquisition of income. Previous empirical studies either have shown that relationship to be negative -work effort decreases with increasing incomes or at the least, have failed to show a positive correlation. The data used in those earlier studies were intercity and interindustry cross-sections [2, 7, 8], industry data over time [71, and cross-sections of occupational sub-groups within a society [2]. The major purpose of this study is to test the conclusions of those earlier investigations to see if these very different international aggregate data confirm that the relation is negative. A secondary objective is to use these data to evaluate possible influences, additional to income, on a society's allocation of effort.

ONE of the principal inputs in the process of producing higher education is past higher education. In order to turn out individuals with university degrees, it is necessary that some of the past recipients of such degrees shall have chosen to join university faculties. The recognition that university graduates are the output of higher education, that faculties are the capital stock, and that the hiring of recent graduates to faculties is the investment process, permits the future growth of higher education in the United States to be analyzed within a Harrod growth framework. Although other concepts are needed even at a high level of abstraction, the capital-output ratio (i.e., the faculty-student ratio) and the investment-output ratio (i.e., the ratio of increments of higher education faculties to past recipients of degrees), are central to this analysis of higher education. The essential difference between this paper and typical growth models is that output in higher education is here treated as a parameter rather than a variable. It is possible that the pressures of rapidly increasing applications to institutions of higher education might result largely in increasing rejections, but the more likely course is the expansion of existing universities and the establishment of new ones.' The purpose of this paper is to show the kinds of pressure and the extent of the pressure which may appear in American higher education as a result of various plausible enrollment rates between now and 1980.2 In the growth model of this paper and the forecasts that result from it will be seen the tremendous strain which the next decade will probably place upon American universities and colleges. What is interesting is not, of course, the existence of this strain long ago realized by educators but the measures of its depth and duration. If the faculty investment rate is not increased, faculty-student ratios will very probably fall by 23% (from .089 to .069) between 1957-1958 and 1967-1968. But the very process of producing this vastly increased amount of higher education produces a greatly increased potential later rise of faculties. After 1967-1968, again if the faculty investment rate remains unchanged, faculty-student ratios will begin to rise almost as dramatically as they fell and will re-attain levels above .08 by 19791980. The strain on faculties during the 1960's tends automatically to reduce this strain in the 1970's and possibly to produce slack thereafter. If faculty salaries adjust to prevent, at least partially, these strains and slacks, faculty pay may nearly double during the next two decades, but the rise will not be smooth. Salaries may rise by 7 % per annum during the 1960's but only by 1% per annum in the 1970's. The financial future of those who profess in higher education may be neither so stable nor so bright as is commonly believed.